20 research outputs found
Pseudo-Diagonals and Uniqueness Theorems
We examine a certain type of abelian C*-subalgebras that allow one to give a
unified treatment of two uniqueness theorems: for graph C*-algebras and for
certain reduced crossed products
Regular ideals, ideal intersections, and quotients
Let be an inclusion of C-algebras. We study the
relationship between the regular ideals of and regular ideals of . We
show that if is a regular C-inclusion and there is a
faithful invariant conditional expectation from onto , then there is an
isomorphism between the lattice of regular ideals of and invariant regular
ideals of . We study properties of inclusions preserved under quotients by
regular ideals. This includes showing that if is a Cartan
inclusion and is a regular ideal in , then is a Cartan
subalgebra of . We provide a description of regular ideals in reduced
crossed products .Comment: 26 pages. Major revision on earlier version of the pape
Cartan subalgebras in C*-algebras of Hausdorff etale groupoids
The reduced -algebra of the interior of the isotropy in any Hausdorff
\'etale groupoid embeds as a -subalgebra of the reduced
-algebra of . We prove that the set of pure states of with unique
extension is dense, and deduce that any representation of the reduced
-algebra of that is injective on is faithful. We prove that there
is a conditional expectation from the reduced -algebra of onto if
and only if the interior of the isotropy in is closed. Using this, we prove
that when the interior of the isotropy is abelian and closed, is a Cartan
subalgebra. We prove that for a large class of groupoids with abelian
isotropy---including all Deaconu--Renault groupoids associated to discrete
abelian groups--- is a maximal abelian subalgebra. In the specific case of
-graph groupoids, we deduce that is always maximal abelian, but show by
example that it is not always Cartan.Comment: 14 pages. v2: Theorem 3.1 in v1 incorrect (thanks to A. Kumjain for
pointing out the error); v2 shows there is a conditional expectation onto
iff the interior of the isotropy is closed. v3: Material (including some
theorem statements) rearranged and shortened. Lemma~3.5 of v2 removed. This
version published in Integral Equations and Operator Theor
INTERMEDIATE C∗-ALGEBRAS OF CARTAN EMBEDDINGS
Let A be a C*-algebra and let D be a Cartan subalgebra of A. We study the following question: if B is a C*-algebra such that D B A, is D a Cartan subalgebra of B? We give a positive answer in two cases: the case when there is a faithful conditional expectation from A onto B, and the case when A is nuclear and D is a C*-diagonal of A. In both cases there is a one-to-one correspondence between the intermediate C*-algebras B, and a class of open subgroupoids of the groupoid G, where ! G is the twist associated with the embedding D A